Looking at the pics again it almost looks like the maker joined. The two pieces of the leaves by just satin stitching the edges and now thru use the edges have wiggled out of the satin stitch.
You say the quilt is old, did you make it or somehow acquire it. Depending on its history you could hand satin stitch those leaves back together just working with that back fab, or as others have suggested cover with ribbon or fab stri.

Have you thought of perhaps using thin bias strips to cover it up ? Just a thought.

Applique a coordinating flat braid over the areas so that it looks like part of the design of the quilt.

Do you have anymore of the fabric? I would pick out the large stipple, make another leaf. Sew the two colors of the leaf with a quarter inch seam down the middle, turn the seam to one side, do not press open. Satin stitch down the center. Lay your leaf down on lightweight interfacing face down and sew around the edge to make the same size pattern after sewn....slit the interfacing and turn inside out. Remove the exiting leaf. Place your new leaf on your quilt top. Hand stitch or tack by machine around the edge. Stipple the leaf down. Is there just one leaf or hundreds on the quilt? If so, satin stitch around the edge and down the middle of the ones that are frayed. Best wishes

It looks like the stitching around the entire leaf is also stressed. I think the easiest is as others mentioned. A ribbon or bias strip that is machine sewn down the center, then a satin stitch around the entire leaf. Good Luck!

I haven't read all the posts but you could use fabric glue to first stop the fraying, then make thin piping strips and glue them down with the fabric glue.

Thanks for all the suggestions! I think I will try the ribbon idea. I can't restitch because this is the backing of an already quilted quilt and it would show on the front. I hope this works!!!

What ever you sew will show on the back or other side. If this would show on the front you will need to hand stitch.
What ever you do on one do on all the shapes to make it a feature not a mistake.

Similar to the suggestions of ribbon, I would use a bias cut tube of fabric (sew right side together, then turn inside out), to create a stem, them stitch it by hand to the middle of the leaf. You could do this to the other leaves or a few of them to make it consistent.